On the Structure of Ideal Class Groups

of CM-Fields

dedicated to Professor K. Kato on his 50th birthday

Masato Kurihara

Received: November 30, 2002 Revised: June 30, 2003

Abstract. For a CM-field K which is abelian over a totally real number field k and a prime number p, we show that the structure of the χ-component A_{K}^{χ} of the p-component of the class group of K is determined by Stickelberger elements (zeta values) (of fields containing K) for an odd character χ of Gal(K/k) satisfying certain conditions. This is a generalization of a theorem of Kolyvagin and Rubin. We define higher Stickelberger ideals using Stickelberger elements, and show that they are equal to the higher Fitting ideals. We also construct and study an Euler system of Gauss sum type for such fields. In the appendix, we determine the initial Fitting ideal of the non-Teichmüller component of the ideal class group of the cyclotomic Z_{p}-extension of a general CM-field which is abelian over k.

0 Introduction

It is well-known that the cyclotomic units give a typical example of Euler systems. Euler systems of this type were systematically investigated by Kato [8], Perrin-Riou [14], and in the book by Rubin [18]. In this paper, we propose to study Euler systems of Gauss sum type which are not Euler systems in the sense of [18]. We construct an Euler system in the multiplicative groups of CM-fields, which is a generalization of the Euler system of Gauss sums, and generalize a structure theorem of Kolyvagin and Rubin for the minus class groups of imaginary abelian fields to general CM-fields. The aim of this paper is to prove the structure theorem (Theorem 0.1 below), and we do not pursue general results on the Euler systems of Gauss sum type in this paper. One of very deep and remarkable works of Kato is his construction of the Euler system (which lies in H^{1}(T)) for a Z_{p}-representation T associated to a modular form. We remark that we do not have an Euler system of Gauss sum type in H^{1}(T), but fixing n > 0 we can find an Euler system of Gauss sum type in H^{1}(T/p^{n}), which will be studied in our forthcoming paper. We will describe our main result. Let k be a totally real number field, and K be a CM-field containing k such that K/k is finite and abelian. We consider an odd prime number p and the p-primary component A_{K} = Cl_{K} ⊗ Z_{p} of the ideal class group of K. Suppose that p does not divide [K : k]. Then, A_{K} is decomposed into A_{K} = ⨁_{χ} A_{K}^{χ} where A_{K}^{χ} is the χ-component which is an O_{χ}-module (where O_{χ} = Z_{p}[Image χ], for the precise definition, see 1.1), and χ ranges over Q_{p}-conjugacy classes of \overline{Q_{p}}^{×}-valued characters of Gal(K/k) (see also 1.1). For k = Q and K = Q(μ_{p}) (the cyclotomic field of p-th roots of unity), Rubin in [17] described the detail of Kolyvagin’s method ([10] Theorem 7), and determined the structure of A_{Q(μ_{p})}^{χ} as a Z_{p}-module for an odd χ, by using the Euler system of Gauss sums (Rubin [17] Theorem 4.4). We generalize this result to arbitrary CM-fields. In our previous paper [11], we proposed a new definition of the Stickelberger ideal. In this paper, for certain CM-fields, we define higher Stickelberger ideals which correspond to higher Fitting ideals. In §3, using the Stickelberger elements of fields containing K, we define the higher Stickelberger ideals Θ_{i,K} ⊂ Z_{p}[Gal(K/k)] for i ≥ 0 (cf. 3.2). Our definition is different from Rubin’s. (Rubin defined the higher Stickelberger ideal using the argument of Euler systems. We do not use the argument of Euler systems to define our Θ_{i,K}.) We remark that our Θ_{i,K} is numerically computable, since the Stickelberger elements are numerically computable. We consider the χ-component Θ_{i,K}^{χ}. We study the structure of the χ-component A_{K}^{χ} as an O_{χ}-module. We note that p is a prime element of O_{χ} because the order of Image χ is prime to p.

Theorem 0.1. We assume that the Iwasawa μ-invariant of K is zero (cf. Proposition 2.1), and χ is an odd character of Gal(K/k) such that χ ≠ ω (where ω is the Teichmüller character giving the action on μ_{p}), and that χ(𝔭) ≠ 1 for every prime 𝔭 of k above p. Suppose that

A_{K}^{χ} ≃ O_{χ}/(p^{n_{1}}) ⊕ ... ⊕ O_{χ}/(p^{n_{r}})

with 0 < n_{1} ≤ ... ≤ n_{r}. Then, for any i with 0 ≤ i < r, we have

(p^{n_{1}+...+n_{r−i}}) = Θ_{i,K}^{χ}

and Θ_{i,K}^{χ} = (1) for i ≥ r. Namely,

A_{K}^{χ} ≃ ⨁_{i≥0} Θ_{i,K}^{χ}/Θ_{i+1,K}^{χ}.

In the case K = Q(μ_{p}) and k = Q, Theorem 0.1 is equivalent to Theorem 4.4 in Rubin [17].

This theorem says that the structure of A_{K}^{χ} as an O_{χ}-module is determined by the Stickelberger elements. Since the Stickelberger elements are defined from the partial zeta functions, we may view our theorem as a manifestation of a very general phenomena in number theory that zeta functions give us information on various important arithmetic objects.

In general, for a commutative ring R and an R-module M such that

R^{m} ⟶^{f} R^{r} ⟶ M ⟶ 0

is an exact sequence of R-modules, the i-th Fitting ideal of M is defined to be the ideal of R generated by all (r−i)×(r−i) minors of the matrix corresponding to f for i with 0 ≤ i < r. If i ≥ r, it is defined to be R. (For more details, see Northcott [13]). Using this terminology, Theorem 0.1 can be simply stated as

Fitt_{i,O_{χ}}(A_{K}^{χ}) = Θ_{i,K}^{χ}

for all i ≥ 0.

The proof of Theorem 0.1 is divided into two parts. We first prove the inclusion Fitt_{i,O_{χ}}(A_{K}^{χ}) ⊃ Θ_{i,K}^{χ}. To do this, we need to consider a general CM-field which contains K. Suppose that F is a CM-field containing K such that F/k is abelian, and F/K is a p-extension. Put R_{F} = Z_{p}[Gal(F/k)]. For a character χ satisfying the conditions in Theorem 0.1, we consider R_{F}^{χ} = O_{χ}[Gal(F/K)] and A_{F}^{χ} = A_{F} ⊗_{R_{F}} R_{F}^{χ} where Gal(K/k) acts on O_{χ} via χ. For the χ-component θ_{F}^{χ} ∈ R_{F}^{χ} of the Stickelberger element of F (cf. 1.2), we do not know whether θ_{F}^{χ} ∈ Fitt_{0,R_{F}^{χ}} (A_{F}^{χ}) always holds or not (cf. Popescu [15] for function fields). But we will show in Corollary 2.4 that the dual version of this statement holds, namely

ι(θ_{F}^{χ}) ∈ Fitt_{0,R_{F}^{χ^{−1}}} ((A_{F}^{χ})^{∨})

where ι : R_{F} ⟶ R_{F} is the map induced by σ ↦ σ^{−1} for σ ∈ Gal(F/k), and (A_{F}^{χ})^{∨} is the Pontrjagin dual of A_{F}^{χ}. We can also determine the right hand side Fitt_{0,R_{F}^{χ^{−1}}} ((A_{F}^{χ})^{∨}). In the Appendix, for the cyclotomic Z_{p}-extension F_{∞}/F,

we determine the initial Fitting ideal of (the Pontrjagin dual of) the nonω component of the p-primary component of the ideal class group of F_{∞} as a Λ_{F} = Z_{p}[[Gal(F_{∞}/k)]]-module (we determine Fitt_{0,Λ_{F}} ((A_{F_{∞}})^{∨}) except ωcomponent, see Theorem A.5). But for the proof of Theorem 0.1, we only need Corollary 2.4 which can be proved more simply than Theorem A.5, so we postpone Theorem A.5 and its proof until the Appendix. Concerning the Iwasawa module X_{F_{∞}} = lim_{←}A_{F_{n}} where F_{n} is the n-th layer of F_{∞}/F, we computed in [11] the initial Fitting ideal under certain hypotheses, for example, if F/Q is abelian. Greither in his recent paper [4] computed the initial Fitting ideal of X_{F_{∞}} more generally. In our previous paper [11] §8, we showed that information on the initial Fitting ideal of the class group of F yields information on the higher Fitting ideals of the class group of K. Using this method, we will show Fitt_{i,O_{χ}}(A_{K}^{χ}) ⊃ Θ_{i,K}^{χ} in Proposition 3.2.

In order to prove the other inclusion, we will use the argument of Euler systems. By Corollary 2.4 which was mentioned above, we obtain

θ_{F}^{χ} A_{F}^{χ} = 0.

(We remark that this has been obtained recently also in Greither [4] Corollary 2.7.) Using this property, we show that for any finite prime ρ of F there is an element g_{F,ρ}^{χ} ∈ (F^{×} ⊗ Z_{p})^{χ} such that div(g_{F,ρ}^{χ}) = θ_{F}^{χ} [ρ]^{χ} in the divisor group where [ρ] is the divisor corresponding to ρ (for the precise relation, see §4). These g_{F,ρ}^{χ}’s become an Euler system of Gauss sum type (see §4). For the Euler system of Gauss sums, a crucial property is Theorem 2.4 in Rubin [18] which is a property on the image in finite fields, and which was proved by Kolyvagin, based on the explicit form of Gauss sums. But we do not know the explicit form of our g_{F,ρ}^{χ}, so we prove, by a completely different method, the corresponding property (Proposition 4.7) which is a key proposition in §4.

It is possible to generalize Theorem 0.1 to characters of order divisible by p satisfying some conditions. We hope to come back to this point in our forthcoming paper.

I would like to express my sincere gratitude to K. Kato for introducing me to the world of arithmetic when I was a student in the 1980’s. It is my great pleasure to dedicate this paper to Kato on the occasion of his 50th birthday. I would like to thank C. Popescu heartily for a valuable discussion on Euler systems. I obtained the idea of studying the elements g_{F,ρ}^{χ} from him. I would also like to thank the referee for his careful reading of this manuscript, and for his pointing out an error in the first version of this paper. I heartily thank C. Greither for sending me his recent preprint [4].

Notation

Throughout this paper, p denotes a fixed odd prime number. We denote by ord_{p} : Q^{×} ⟶ Z the normalized discrete valuation at p. For a positive integer n, μ_{n} denotes the group of all n-th roots of unity. For a number field F, O_{F} denotes the ring of integers. For a group G and a G-module M, M^{G} denotes the G-invariant part of M (the maximal subgroup of M on which G acts trivially), and M_{G} denotes the G-coinvariant of M (the maximal quotient of M on which G acts trivially). For a commutative ring R, R^{×} denotes the unit group.

1 Preliminaries

1.1. Let G be a profinite abelian group such that G = Δ×G′ where #Δ is finite and prime to p, and G′ is a pro-p group. We consider the completed group ring Z_{p}[[G]] which is decomposed into

Z_{p}[[G]] = Z_{p}[Δ][[G′]] ≃ ⨁_{χ} O_{χ}[[G′]]

where χ ranges over all representatives of Q_{p}-conjugacy classes of characters of Δ (a \overline{Q_{p}}^{×}-valued character χ is said to be Q_{p}-conjugate to χ′ if σχ = χ′ for some σ ∈ Gal(\overline{Q_{p}}/Q_{p})), and O_{χ} is Z_{p}[Image χ] as a Z_{p}-module, and Δ acts on it via χ (σx = χ(σ)x for σ ∈ Δ and x ∈ O_{χ}). Hence, any Z_{p}[[G]]-module M is decomposed into M ≃ ⨁_{χ} M^{χ} where

M^{χ} ≃ M ⊗_{Z_{p}[Δ]} O_{χ} ≃ M ⊗_{Z_{p}[[G]]} O_{χ}[[G′]].

In particular, M^{χ} is an O_{χ}[[G′]]-module. For an element x of M, the χcomponent of x is denoted by x^{χ} ∈ M^{χ}. Let 1_{Δ} be the trivial character σ ↦ 1 of Δ. We denote by M^{1} the trivial character component, and define M^{∗} to be the component obtained from M by removing M^{1}, namely

M = M^{1} ⊕ M^{∗}.

Suppose further that G′ = G × G′′ where G is a finite p-group. Let ψ be a character of G. We regard χψ as a character of G_{0} = Δ × G, and define M^{χψ} by M^{χψ} = M ⊗_{Z_{p}[G_{0}]} O_{χψ} where O_{χψ} is O_{χψ} = Z_{p}[Image χψ] on which G_{0} acts via χψ. By definition, if χ ≠ 1_{Δ}, we have M^{χψ} ≃ (M^{∗})^{χψ}.

Let k be a totally real number field and F be a CM-field such that F/k is finite and abelian, and μ_{p} ⊂ F. We denote by F_{∞}/F the cyclotomic Z_{p}-extension, and put G = Gal(F_{∞}/k). We write G = Δ × G′ as above. A Z_{p}[[G]]-module M is decomposed into M = M^{+} ⊕ M^{−} with respect to the action of the complex conjugation where M^{±} is the ±-eigenspace. By definition, M^{−} = ⨁_{χ:odd} M^{χ} where χ ranges over all odd characters of Δ. We consider the Teichmüller character ω giving the action of Δ on μ_{p}, and define M^{∼} to be the component obtained from M^{−} by removing M^{ω}, namely

M^{−} = M^{∼} ⊕ M^{ω}.

For an element x of M, we write x^{∼} the component of x in M^{∼}.

1.2. Let k, F, F_{∞} be as in 1.1, and S be a finite set of finite primes of k containing all the primes which ramify in F/k. We define in the usual way the partial zeta function for σ ∈ Gal(F/k) by

ζ_{S}(s, σ) = ∑_{(𝔞,F/k)=σ 𝔞 is prime to S} N(𝔞)^{−s}

for Re(s) > 1 where N(𝔞) is the norm of 𝔞, and 𝔞 runs over all integral ideals of k, coprime to the primes in S such that the Artin symbol (𝔞, F/k) is equal to σ. The partial zeta functions are meromorphically continued to the whole complex plane, and holomorphic everywhere except for s = 1. We define

θ_{F,S} = ∑_{σ∈Gal(F/k)} ζ_{S}(0, σ)σ^{−1}

which is an element of Q[Gal(F/k)] (cf. Siegel [21]). Suppose that S_{F} is the set of ramifying primes of k in F/k. We simply write θ_{F} for θ_{F,S_{F}}. We know by Deligne and Ribet the non ω-component (θ_{F,S})^{∼} ∈ Q_{p}[Gal(F/k)]^{∼} is in Z_{p}[Gal(F/k)]^{∼}. In particular, for a character χ of Δ with χ ≠ ω, we have (θ_{F,S})^{χ} ∈ Z_{p}[Gal(F/k)]^{χ}. Suppose that S contains all primes above p. Let F_{n} denote the n-th layer of the cyclotomic Z_{p}-extension F_{∞}/F, and consider (θ_{F_{n},S})^{∼} ∈ Z_{p}[Gal(F_{n}/k)]^{∼}. These θ_{F_{n},S}^{∼}’s become a projective system with respect to the canonical restriction maps, and we define

θ_{F_{∞},S}^{∼} ∈ Z_{p}[[Gal(F_{∞}/k)]]^{∼}

to be their projective limit. This is essentially (the non ω-part of) the p-adic L-function of Deligne and Ribet [1].

2 Initial Fitting ideals

Let k, F, F_{∞} be as in §1. We denote by k_{∞}/k the cyclotomic Z_{p}-extension, and assume that F ∩ k_{∞} = k. Our aim in this section is to prove Proposition 2.1 and Corollary 2.4 below.

2.1. Let S be a finite set of finite primes of k containing ramifying primes in F_{∞}/k. We denote by F^{+} the maximal real subfield of F. Put Λ_{F} = Z_{p}[[Gal(F_{∞}/k)]] and Λ_{F^{+}} = Z_{p}[[Gal(F_{∞}^{+}/k)]] which is naturally isomorphic to the plus part Λ_{F}^{+} of Λ_{F}. We denote by M_{∞,S} the maximal abelian pro-p extension of F_{∞}^{+} which is unramified outside S, and by X_{F_{∞}^{+},S} the Galois group of M_{∞,S}/F_{∞}^{+}. We study X_{F_{∞}^{+},S} which is a torsion Λ_{F^{+}}-module. We consider a ring homomorphism τ^{−1}ι : Λ_{F} ⟶ Λ_{F} which is defined by σ ↦ κ(σ)σ^{−1} for σ ∈ Gal(F_{∞}/k) where κ : Gal(F_{∞}/k) ⟶ Z_{p}^{×} is the cyclotomic character giving the action of Gal(F_{∞}/k) on μ_{p^{∞}}. Let (Λ_{F})^{∼} and (Λ_{F}^{+})^{∗} = (Λ_{F^{+}})^{∗} be as in §1.1. Then, τ^{−1}ι induces

τ^{−1}ι : (Λ_{F})^{∼} ⟶ (Λ_{F^{+}})^{∗}.

Let θ_{F_{∞},S}^{∼} ∈ (Λ_{F})^{∼} be the Stickelberger element defined in 1.2.

Proposition 2.1. Assume that the the Iwasawa μ-invariant of F is zero, namely X_{F_{∞}^{+},S} is a finitely generated Z_{p}-module. Then, Fitt_{0,Λ_{F^{+}}} ((X_{F_{∞}^{+},S})^{∗}) is generated by τ^{−1}ι(θ_{F_{∞,S}}^{∼}) except the trivial character component, namely

Fitt_{0,Λ_{F^{+}}} ((X_{F_{∞}^{+},S})^{∗})^{∗} = (τ^{−1}ι(θ_{F_{∞},S}^{∼})).

Proof. We use the method in [11]. In fact, the proof of this proposition is much easier than that of Theorem 0.9 in [11]. We decompose G = Gal(F_{∞}/k) as in 1.1 (G = Δ × G′). Suppose that c is the complex conjugation in Δ and put Δ^{+} = Δ/ < c >, and G_{0} = Gal(F^{+}/k). Then, we can write G_{0} = Δ^{+} × G where G is a p-group. For a character χ of Δ^{+} with χ ≠ 1_{Δ^{+}}, and a character ψ of G, we regard χψ as a character of G_{0}. We consider (X_{F_{∞}^{+},S})^{χψ} which is an O_{χψ}[[Gal(F_{∞}/F)]]-module (cf. 1.1). Our assumption of the vanishing of the μ-invariant implies that (X_{F_{∞}^{+},S})^{χψ} is a finitely generated O_{χψ}-module. We will first show that (X_{F_{∞}^{+},S})^{χψ} is a free O_{χψ}-module. Let H ⊂ G be the kernel of ψ, and M be the subfield of F corresponding to H, namely Gal(F/M) = H. We denote by M_{∞} the cyclotomic Z_{p}- extension of M and regard H as the Galois group of F_{∞}/M_{∞}. We will see that the H-coinvariant ((X_{F_{∞}^{+},S})^{χ})_{H} is naturally isomorphic to (X_{M_{∞}^{+},S})^{χ}. In fact, by taking the dual, it is enough to show that the natural map H_{et}^{1}(O_{M_{∞}^{+}}[1/S], Q_{p}/Z_{p})^{χ^{−1}} ⟶ (H_{et}^{1}(O_{F_{∞}^{+}}[1/S], Q_{p}/Z_{p})^{χ^{−1}})^{H} of etale cohomology groups is bijective where O_{M_{∞}^{+}}[1/S] (resp. O_{F_{∞}^{+}}[1/S]) is the ring of S-integers in M_{∞}^{+} (resp. F_{∞}^{+}). This follows from the Hochschild-Serre spectral sequence and H^{1}(H, Q_{p}/Z_{p})^{χ^{−1}} = H^{2}(H, Q_{p}/Z_{p})^{χ^{−1}} = 0. Hence, regarding χψ as a character of Gal(M^{+}/k), we have

(X_{F_{∞}^{+},S})^{χψ} = (X_{M_{∞}^{+},S})^{χψ}.

We note that (X_{M_{∞}^{+},S})^{χ} does not have a nontrivial finite O_{χ}[[G′]]-submodule (Theorem 18 in Iwasawa [5]), so is free over O_{χ} by our assumption of the μinvariant. We will use the same method as Lemma 5.5 in [11] to prove that (X_{M_{∞}^{+},S})^{χψ} is free over O_{χψ}. We may assume ψ ≠ 1_{G}, so p divides the order of Gal(M^{+}/k). Let C be the subgroup of Gal(M^{+}/k) of order p, M′ the subfield such that Gal(M^{+}/M′) = C, and put N_{C} = Σ_{σ∈C}σ. We have an isomorphism (X_{M_{∞}^{+},S})^{χψ} ≃ (X_{M_{∞}^{+},S})^{χ}/(N_{C}). Let σ_{0} be a generator of C. In order to prove that (X_{M_{∞}^{+},S})^{χψ} is free over O_{χψ}, it is enough to show that the map

σ_{0} − 1 : (X_{M_{∞}^{+},S})^{χ}/(N_{C}) ⟶ (X_{M_{∞}^{+},S})^{χ}

is injective. Hence, it suffices to show ((X_{M_{∞}^{+},S})^{χ})^{C} = N_{C}((X_{M_{∞}^{+},S})^{χ}), hence to show Ĥ^{0}(C, (X_{M_{∞}^{+},S})^{χ}) = 0. Taking the dual, it is enough to show H^{1}(C, H_{et}^{1}(O_{M_{∞}^{+}}[1/S], Q_{p}/Z_{p})^{χ^{−1}}) = 0. This follows from the Hochschild-Serre spectral sequence and H_{et}^{2}(O_{M′_{∞}}[1/S], Q_{p}/Z_{p}) = 0 (which is a famous property called the weak Leopoldt conjecture and which follows immediately from the vanishing of the p-component of the Brauer group of M′_{∞}). Thus, (X_{F_{∞}^{+},S})^{χψ} is a free O_{χψ}-module of finite rank. This shows that Fitt_{0,O_{χψ}[[Gal(F_{∞}/F)]]}((X_{F_{∞}^{+},S})^{χψ}) coincides with its characteristic ideal. By Wiles [25] and our assumption, the μ-invariant of (τ^{−1}ι(θ_{F_{∞},S}^{∼}))^{χψ} is also zero, and by the main conjecture proved by Wiles [25], we have

Fitt_{0,O_{χψ}[[Gal(F_{∞}/F)]]}((X_{F_{∞}^{+},S})^{χψ}) = (τ^{−1}ι(θ_{F_{∞},S}^{∼})^{χψ}).

This holds for any χ and ψ with χ ≠ 1_{Δ}. Hence, by Corollary 4.2 in [11], we obtain the conclusion of Proposition 2.1.

2.2. For any number field F, we denote by A_{F} the p-primary component of the ideal class group of F. Let F be as above. We define

A_{F_{∞}} = lim_{→}A_{F_{n}}

where F_{n} is the n-th layer of F_{∞}/F. We denote by (A_{F_{∞}})^{∨} the Pontrjagin dual of A_{F_{∞}}. Let S_{p} be the set of primes of k lying over p. By the orthogonal pairing in P.276 of Iwasawa [5] which is defined by the Kummer pairing, we have an isomorphism

(X_{F_{∞,Sp}^{+}} )^{∗} ≃ (A_{F_{∞}}^{∼})^{∨}(1).

Let ι : Λ_{F} ⟶ Λ_{F} be the ring homomorphism induced by σ ↦ σ^{−1} for σ ∈ Gal(F_{∞}/k). For a character χ of Δ, ι induces a ring homomorphism Λ_{F}^{χ} ⟶ Λ_{F}^{χ^{−1}} which we also denote by ι. Since there is a natural surjective homomorphism (X_{F_{∞,S}^{+}})^{∗} ⟶ (X_{F_{∞,Sp}^{+}} )^{∗}, Proposition 2.1 together with the above isomorphism implies

Corollary 2.2. Let χ be an odd character of Δ such that χ ≠ ω. Under the assumption of Proposition 2.1, we have

ι(θ_{F_{∞},S}^{χ}) ∈ Fitt_{0,Λ_{F}^{χ^{−1}}} ((A_{F_{∞}}^{χ})^{∨}).

Next, we consider a general CM-field F such that F/k is finite and abelian (Here, we do not assume μ_{p} ⊂ F). Put R_{F} = Z_{p}[Gal(F/k)]. Let G be the pprimary component of Gal(F/k), and Gal(F/k) = Δ×G. Suppose that χ is an odd character of Δ with χ ≠ ω. We consider R_{F}^{χ} = O_{χ}[G], and define ι : R_{F} ⟶ R_{F} and ι : R_{F}^{χ} ⟶ R_{F}^{χ^{−1}} similarly as above. If we assume that the Iwasawa μinvariant of F vanishes, (X_{F (μ_{p})_{∞},S})^{χ^{−1}ω} is a finitely generated O_{χ}-module, so we can apply the proof of Proposition 2.1 to get ι(θ_{F_{∞},S}^{χ}) ∈ Fitt_{0,Λ_{F}^{χ^{−1}}} ((A_{F_{∞}}^{χ})^{∨}).

Since A_{F}^{−} ⟶ A_{F (μ_{p})_{∞}}^{−} is injective ([24] Prop.13.26), (A_{F (μ_{p})_{∞}}^{χ})^{∨} ⟶ (A_{F}^{χ})^{∨} is

surjective. The image of ι(θ_{F_{∞},S}^{χ}) ∈ Λ_{F}^{χ^{−1}} in R_{F}^{χ^{−1}} is ι(θ_{F,S}^{χ}). Hence, we obtain Corollary 2.3. Assume that the Iwasawa μ-invariant of F is zero. Then, we have

ι(θ_{F,S}^{χ}) ∈ Fitt_{0,R_{F}^{χ^{−1}}} ((A_{F}^{χ})^{∨}).

Let S_{F (μ_{p})_{∞}} (resp. S_{F}) be the set of ramifying primes in F (μ_{p})_{∞}/k (resp. F/k). Note that S_{F (μ_{p})_{∞}} \ S_{F} ⊂ S_{p} and

θ_{F,S_{F (μp)∞}} = (Π_{𝔭∈S_{F (μp)∞} \S_{F}} (1 − φ_{𝔭}^{−1}))θ_{F,S_{F}}

where φ_{𝔭} is the Frobenius of 𝔭 in Gal(F/k). If χ(𝔭) ≠ 1 for all 𝔭 above p, (1 − φ_{𝔭}^{−1})^{χ} is a unit of R_{F}^{χ^{−1}} because the order of χ is prime to p. Therefore, we get

Corollary 2.4. Assume that the Iwasawa μ-invariant of F is zero, and that χ(𝔭) ≠ 1 for all 𝔭 above p. Then, we have

ι(θ_{F}^{χ}) ∈ Fitt_{0,R_{F}^{χ^{−1}}} ((A_{F}^{χ})^{∨}).

3 Higher Stickelberger ideals

In this section, for a finite abelian extension K/k whose degree is prime to p, we will define the ideal Θ_{i,K} ⊂ Z_{p}[Gal(K/k)] for i ≥ 0. We also prove the inclusion Θ_{i,K}^{χ} ⊂ Fitt_{i,O_{χ}}(A_{K}^{χ}) for K and χ as in Theorem 0.1.

3.1. In this subsection, we assume that O is a discrete valuation ring with maximal ideal (p). We denote by ord_{p} the normalized discrete valuation of O, so ord_{p}(p) = 1. For n, r > 0, we consider a ring

A_{n,r} = O[[S_{1}, ..., S_{r}]]/((1 + S_{1})^{p^{n}} − 1, ..., (1 + S_{r})^{p^{n}} − 1).

Suppose that f is an element of A_{n,r} and write f = Σ_{i_{1},...i_{r}≥0}a_{i_{1}...i_{r}}S_{1}^{i_{1}}...S_{r}^{i_{r}} mod I where I = ((1+S_{1})^{p^{n}} −1, ..., (1+S_{r})^{p^{n}} −1). For positive integers i and s, we set s′ = min{x ∈ Z : s < p^{x}}. Assume s′ ≤ n. If 0 < j < p^{s′}, we have ord_{p}(\binom{p^{n}}{j}) = ord_{p}(p^{n}!/(j!(p^{n} − j)!)) ≥ n − s′ + 1. Hence, for i_{1}, ..., i_{r} ≤ s < p^{s′}, a_{i_{1},...,i_{r}} mod p^{n−s′+1} is well-defined from f ∈ A_{n,r}. For positive integers i and s with s′ ≤ n, we define I_{i,s}(f) to be the ideal of O which is generated by p^{n−s′+1} and

{a_{i_{1},...,i_{r}} : 0 ≤ i_{1}, ..., i_{r} ≤ s and i_{1} + ... + i_{r} ≤ i}.

Since a_{i_{1},...,i_{r}} is well-defined mod p^{n−s′+1}, I_{i,s}(f) ⊂ O is well-defined for any i and s ∈ Z_{>0} such that n ≥ s′.

Lemma 3.1. Let α : A_{n,r} ⟶ A_{n,r} be the homomorphism of O-algebras defined by α(S_{k}) = Π_{j=1}^{r}(1+S_{j})^{a_{kj}} −1 for 1 ≤ k, j ≤ r such that (a_{kj}) ∈ GL_{n}(Z/p^{n}Z). Then, we have

I_{i,s}(α(f)) = I_{i,s}(f).

Proof. It is enough to show I_{i,s}(f) ⊂ I_{i,s}(α(f)) because if we obtain this inclusion, the other inclusion is also obtained by applying it to α^{−1}. Further, since (a_{kj}) is a product of elementary matrices, it suffices to show the inclusion in the case that α corresponds to an elementary matrix, in which case, the inclusion can be easily checked.

In particular, let ι : A_{n,r} ⟶ A_{n,r} be the ring homomorphism defined by ι(S_{k}) = (1 + S_{k})^{−1} − 1 for k = 1, ..., r. Then, we have

I_{i,s}(ι(f)) = I_{i,s}(f)

which we will use later.

3.2. Suppose that k is totally real, K is a CM-field, and K/k is abelian such that p does not divide [K : k]. Put Δ = Gal(K/k). For i ≥ 0, we will define the higher Stickelberger ideal Θ_{i,K} ⊂ Z_{p}[Δ]. Since Z_{p}[Δ] ≃ ⨁_{χ} O_{χ}, it is enough to define (Θ_{i,K})^{χ}. We replace K by the subfield corresponding to the kernel of χ, and suppose the conductor of K/k is equal to that of χ. For n, r > 0, let S_{K,n,r} denote the set of CM fields F such that K ⊂ F, F/k is abelian, and F/K is a p-extension satisfying Gal(F/K) ≃ (Z/p^{n})^{⊕r}. For F ∈ S_{K,n,r}, we have an isomorphism

Z_{p}[Gal(F/k)]^{χ} ≃ Z_{p}[Δ]^{χ}[Gal(F/K)] = O_{χ}[Gal(F/K)].

Fixing generators of Gal(F/K), we have an isomorphism between O_{χ}[Gal(F/K)] and A_{n,r} with O = O_{χ} in 3.1 (the fixed generators σ_{1},...,σ_{r} correspond to 1 + S_{1},...,1 + S_{r}). We first assume χ is odd and χ ≠ ω. Then, θ_{F}^{χ} is in Z_{p}[Gal(F/k)]^{χ} = O_{χ}[Gal(F/K)] (cf. 1.2). Using the isomorphism between O_{χ}[Gal(F/K)] and A_{n,r}, for i and s such that n ≥ s′, we define the ideal I_{i,s}(θ_{F}^{χ}) of O_{χ} (cf. 3.1). By Lemma 3.1, I_{i,s}(θ_{F}^{χ}) does not depend on the choice of generators of Gal(F/K). We define (Θ_{0,K})^{χ} = (θ_{K}^{χ}). Suppose that (Θ_{0,K})^{χ} = (p^{m}). If m = 0, we define (Θ_{i,K})^{χ} = (1) for all i ≥ 0. We assume m > 0. We define S_{K,n} = ⋃_{r>0} S_{K,n,r}. We define (Θ_{i,s,K})^{χ} to be the ideal generated by all I_{i,s}(θ_{F}^{χ})’s where F ranges over all fields in S_{K,n} for all n ≥ m + s′ − 1 where s′ = min{x ∈ Z : s < p^{x}} as in 3.1, namely

(Θ_{i,s,K})^{χ} = ⋃_{{F ∈S_{K,n} \atop n≥m+s′−1}} I_{i,s}(θ_{F}^{χ}).

We define (Θ_{i,K})^{χ} by (Θ_{i,K})^{χ} = ⋃_{s>0}(Θ_{i,s,K})^{χ}. For χ satisfying the condition of Theorem 0.1, we will see later in §5 that (Θ_{i,K})^{χ} = (Θ_{i,1,K})^{χ}.

For F ∈ S_{K,m} with m > 0, I_{i,1}(θ_{F}^{χ}) contains p^{m} (note that s′ = 1 when s = 1), so p^{m} ∈ (Θ_{i,K})^{χ}. Since (Θ_{0,K})^{χ} = (p^{m}), (Θ_{0,K})^{χ} is in (Θ_{i,K})^{χ}. It is also clear from definition that (Θ_{i,s,K})^{χ} ⊂ (Θ_{i+1,s,K})^{χ} for i > 0 and s > 0. Hence, we have a sequence of ideals

(Θ_{0,K})^{χ} ⊂ (Θ_{1,K})^{χ} ⊂ (Θ_{2,K})^{χ} ⊂ ...

We do not use the ω-component in this paper, but for χ = ω, we define (Θ_{0,K})^{χ} = (θ_{K} Ann_{Z_{p}[Gal(K/k)]}(μ_{p^{∞}}(K)))^{χ}. For i > 0, (Θ_{i,K})^{χ} is defined similarly as above by using xθ_{F}^{χ} instead of θ_{F}^{χ} where x ranges over elements of Ann_{Z_{p}[Gal(F/k)]}(μ_{p^{∞}}(F))^{χ}. For an even χ, we define (Θ_{i,K})^{χ} = (0) for all i ≥ 0.

Proposition 3.2. Suppose that K and χ be as in Theorem 0.1. Then, for any i ≥ 0, we have

(Θ_{i,K})^{χ} ⊂ Fitt_{i,O_{χ}}(A_{K}^{χ}).

Proof. At first, by Theorem 3 in Wiles [26] we know #A_{K}^{χ} = #(O_{χ}/(θ_{K}^{χ})), hence (Θ_{0,K})^{χ} = Fitt_{0,O_{χ}}(A_{K}^{χ}). (In our case, this is a direct consequence of the main conjecture proved by Wiles [25].) We assume i > 0. By the definition of (Θ_{i,K})^{χ}, we have to show I_{i,s}(θ_{F}^{χ}) ⊂ Fitt_{i,O_{χ}}(A_{K}^{χ}) for F ∈ S_{K,n,r} where the notation is the same as above. By Lemma 3.1, I_{i,s}(θ_{F}^{χ}) = I_{i,s}(ι(θ_{F}^{χ})). Hence, it is enough to show

I_{i,s}(ι(θ_{F}^{χ})) ⊂ Fitt_{i,O_{χ}}(A_{K}^{χ}).

We will prove this inclusion by the same method as Theorem 8.1 in [11]. We write O = O_{χ} = O_{χ^{−1}}, and G = Gal(F/K). As in 3.2, we fix an isomorphism O[Gal(F/K)] ≃ A_{n,r} by fixing generators of G. We consider (A_{F}^{χ})^{∨} = (A_{F} ⊗_{Z_{p}[Δ]} O)^{∨} = (A_{F} ⊗_{Z_{p}[Gal(F/k)]} O[G])^{∨} which is an O[G]-module. Since F/K is a p-extension, it is well-known that the vanishing of the Iwasawa μ-invariant of K implies the vanishing of the Iwasawa μ-invariant of F ([6] Theorem 3). By Corollary 2.4, we have

ι(θ_{F}^{χ}) ∈ Fitt_{0,O[G]}((A_{F}^{χ})^{∨}).

Since χ ≠ ω and χ is odd, for a unit group O_{F}^{×}, we have (O_{F}^{×} ⊗ Z_{p})^{χ} = μ_{p^{∞}}(F)^{χ} = 0, so H^{1}(Gal(F/K), O_{F}^{×})^{χ} = H^{1}(Gal(F/K), (O_{F}^{×} ⊗ Z_{p})^{χ}) = 0. This shows that the natural map A_{K}^{χ} ⟶ A_{F}^{χ} is injective. Hence, regarding A_{K}^{χ} as an O[G]-module (G acting trivially on it), we have

Fitt_{0,O[G]}((A_{F}^{χ})^{∨}) ⊂ Fitt_{0,O[G]}((A_{K}^{χ})^{∨}),

and

ι(θ_{F}^{χ}) ∈ Fitt_{0,O[G]}((A_{K}^{χ})^{∨}).

Hence, by the lemma below, we obtain

I_{i,s}(ι(θ_{F}^{χ})) ⊂ Fitt_{i,O}(A_{K}^{χ}).

This completes the proof of Proposition 3.2.

Lemma 3.3. Put I_{G} = (S_{1}, ..., S_{r}). Then, Fitt_{0,O[G]}((A_{K}^{χ})^{∨}) is generated by Fitt_{j,O}(A_{K}^{χ})(I_{G})^{j} for all j ≥ 0.

Proof. Put M = (A_{K}^{χ})^{∨}. Since O is a discrete valuation ring, M^{∨} is isomorphic to M as an O-module. Hence, Fitt_{j,O}(M) = Fitt_{j,O}(M^{∨}) = Fitt_{j,O}(A_{K}^{χ}). We take generators e_{1},...,e_{m} and relations Σ_{k=1}^{m}a_{kl}e_{k} = 0 (a_{kl} ∈ O, l = 1, 2, ..., m) of M as an O-module. Put A = (a_{kl}). We also consider a relation matrix of M as an O[G]-module. By definition, I_{G} annihilates M. Hence, the relation matrix of M as an O[G]-module is of the form

( \begin{matrix}S_{1} & ... & S_{r} & ... & ... & 0 & ... & 0 \\ 0 & ... & 0 & ... & ... & 0 & ... & 0 \\ . & ... & . & ... & ... & . & ... & . & A \\ . & ... & . & ... & ... & . & ... & . \\ 0 & ... & 0 & ... & ... & S_{1} & ... & S_{r} \end{matrix}  ) .

Therefore, Fitt_{0,O[G]}(M) is generated by Fitt_{j,O}(M)(I_{G})^{j} for all j ≥ 0.

4 Euler systems

Let K/k be a finite and abelian extension of degree prime to p. We also assume that K is a CM-field, and the Iwasawa μ-invariant of K is zero. We consider a CM field F such that F/k is finite and abelian, F ⊃ K, and F/K is a pextension. Since the Iwasawa μ-invariant of F is also zero, by Corollary 2.4, we have ι(θ_{F}^{∼})(A_{F}^{∼})^{∨} = 0. Hence, we have

θ_{F}^{∼}A_{F}^{∼} = 0.

We denote by O_{F}^{×}, Div_{F}, and A_{F} the unit group of F, the divisor group of F, and the p-primary component of the ideal class group of F. We write [ρ] for the divisor corresponding to a finite prime ρ, and write an element of Div_{F} of the form Σa_{i}[ρ_{i}] with a_{i} ∈ Z. If (x) = Πρ_{i}^{a_{i}} is the prime decomposition of x ∈ F^{×}, we write div(x) = Σa_{i}[ρ_{i}] ∈ Div_{F}. Consider an exact sequence 0 ⟶ O_{F}^{×} ⊗ Z_{p} ⟶ F^{×} ⊗ Z_{p} ⟶^{div} Div_{F} ⊗Z_{p} ⟶ A_{F} ⟶ 0. Since the functor M ↦ M^{∼} is exact and (O_{F}^{×} ⊗ Z_{p})^{∼} = 0,

0 ⟶ (F^{×} ⊗ Z_{p})^{∼} ⟶^{div} (Div_{F} ⊗Z_{p})^{∼} ⟶ A_{F}^{∼} ⟶ 0

is exact. For any finite prime ρ_{F} of F, since the class of θ_{F}^{∼}[ρ_{F}]^{∼} in A_{F}^{∼} vanishes, there is a unique element g_{F,ρ_{F}} in (F^{×} ⊗ Z_{p})^{∼} such that

div(g_{F,ρ_{F}}) = θ_{F}^{∼}[ρ_{F}]^{∼}.

By this property, we have Lemma 4.1. Suppose that M is an intermediate field of F/K, and S_{F} (resp. S_{M}) denotes the set of ramifying primes of k in F/k (resp. M/k). Let ρ_{M} be a prime of M, ρ_{F} be a prime of F above ρ_{M}, and f = [O_{F} /ρ_{F} : O_{F} /ρ_{M}]. Then, we have

N_{F/M}(g_{F,ρ_{F}}) = ( ∏_{λ∈S_{F} \S_{M}}(1 − φ_{λ}^{−1}))^{∼}(g_{M,ρ_{M}})^{f}

where N_{F/M} : F^{×} ⟶ M^{×} is the norm map, and φ_{λ} is the Frobenius of λ in Gal(M/k).

Proof. In fact, we have

div(N_{L/M}(g_{F,ρ_{F}})) = c_{F/M}(θ_{F})^{∼}[N_{F/M}(ρ_{F})]^{∼}

where c_{F/M} : Z_{p}[Gal(F/k)] ⟶ Z_{p}[Gal(M/k)] is the map induced by the restriction σ ↦ σ_{|M} and N_{F/M}(ρ_{F}) is the norm of ρ_{F}. By a famous property of the Stickelberger elements (see Tate [23] p.86), we have

c_{F/M}(θ_{F}^{∼}) = (( ∏_{λ∈S_{F} \S_{M}}(1 − φ_{λ}^{−1}))θ_{M})^{∼},

hence the right hand side of the first equation is equal to ((∏_{λ∈S_{F} \S_{M}} (1 − φ_{λ}^{−1}))θ_{M})^{∼}f[ρ_{M}]^{∼}. This is also equal to div((∏_{λ∈S_{F} \S_{M}} (1−φ_{λ}^{−1}))^{∼}(g_{M,ρ_{M}})^{f}). Since div is injective, we get this lemma.

Remark 4.2. By the property θ_{F}^{∼}A_{F}^{∼} = 0, we can also obtain an Euler system in some cohomology groups by the method of Rubin in [18] Chapter 3, section 3.4. But here, we consider the Euler system of these g_{F,ρ_{F}} ’s, which is an analogue of the Euler system of Gauss sums. I obtained the idea of studying the elements g_{F,ρ_{F}} from C. Popescu through a discussion with him.

Let H_{k} be the Hilbert p-class field of k, namely the maximal abelian p-extension of k which is unramified everywhere. Suppose that the p-primary component A_{k} of the ideal class group of k is decomposed into A_{k} = Z/p^{a_{1}}Z⊕...⊕Z/p^{a_{s}}Z. We take and fix a prime ideal 𝔮_{j} which generates the j-th direct summand for each j = 1, ..., s. We take ξ_{j} ∈ k^{×} such that 𝔮_{j}^{p^{a_{j}}} = (ξ_{j}) for each j. Let U denote the subgroup of k^{×} generated by the unit group O_{k}^{×} and ξ_{1},...,ξ_{s}. For a positive integer n > 0, we define P_{n} to be the set of primes of k with degree 1 which are prime to p𝔮_{1} · ... · 𝔮_{s}, and which split completely in KH_{k}(μ_{p^{n}}, U^{1/p^{n}}).

Lemma 4.3. Suppose λ ∈ P_{n}. Then, there exists a unique cyclic extension k_{n}(λ)/k of degree p^{n}, which is unramified outside λ, and in which λ is totally ramified.

Proof. We prove this lemma by class field theory. Let C_{k} (resp. Cl_{k}) be the idele class group (resp. the ideal class group) of k. For a prime v, we denote by k_{v} the completion of k at v, and define U_{k_{v}} to be the unit group of the ring of integers of k_{v} for a finite prime v, and U_{k_{v}} = k_{v} for an infinite prime v. We denote by U_{k_{v}}^{1} the group of principal units for a finite prime v. We define C_{k,λ,n} which is a quotient of C_{k} ⊗ Z_{p} by

C_{k,λ,n} = ((k_{λ}^{×}/U_{k_{λ}}^{1}) ⊗ Z/p^{n}Z ⊕ ⨁_{v≠λ}(k_{v}^{×}/U_{k_{v}}) ⊗ Z_{p})/(the image of k^{×})

where v ranges over all primes except λ. Since λ splits in H_{k}, the class of λ in Cl_{k} ⊗ Z_{p} = A_{k} is trivial. Hence, the natural map

⨁_{v} k_{v}^{×} ⊗ Z_{p} ⟶ ⨁_{v}(k_{v}^{×}/U_{k_{v}}) ⊗ Z_{p} ⟶ (⨁_{v} Z_{p})/(the image of k^{×}) = A_{k}

(v ranges over all primes) induces C_{k,λ,n} ⟶ A_{k}, and we have an exact sequence

(U_{k_{λ}}/U_{k_{λ}}^{1}) ⊗ Z/p^{n}Z ⟶^{a} C_{k,λ,n} ⟶^{b} A_{k} ⟶ 0.

Let κ(λ) denote the residue field of λ. Since λ splits in k(μ_{p^{n}}), (U_{k_{λ}}/U_{k_{λ}}^{1}) ⊗ Z/p^{n}Z = κ(λ)^{×}⊗Z/p^{n}Z is cyclic of order p^{n}. Since λ splits in k(μ_{p^{n}}, (O_{k}^{×})^{1/p^{n}}), O_{k}^{×} is in (U_{k_{λ}})^{p^{n}} and a is injective (Rubin [18] Lemma 4.1.2 (i)). Next, we will show that the exact sequence

0 ⟶ (U_{k_{λ}}/U_{k_{λ}}^{1}) ⊗ Z/p^{n}Z ⟶^{a} C_{k,λ,n} ⟶^{b} A_{k} ⟶ 0

splits. Let 𝔮_{j}, a_{j}, ξ_{j} be as above. Suppose that π_{𝔮_{j}} is a uniformizer of k_{𝔮_{j}}. We denote by Π_{𝔮_{j}} the idele whose 𝔮_{j}-component is π_{𝔮_{j}} and whose v-component is 1 for every prime v except for 𝔮_{j} (the λ-component is also 1). Since λ splits in k(ξ_{j}^{1/p^{n}}), we have ξ_{j} ∈ (U_{k_{λ}})^{p^{n}}. Hence, the class of ξ_{j} ∈ k^{×} in (k_{λ}^{×}/U_{k_{λ}}^{1}) ⊗ Z/p^{n}Z⊕⨁_{v≠λ}(k_{v}^{×}/U_{k_{v}})⊗Z_{p} coincides with (Π_{𝔮_{j}})^{p^{a_{j}}}. This shows that the class [Π_{𝔮_{j}}]_{C_{k,λ,n}} of Π_{𝔮_{j}} in C_{k,λ,n} has order p^{a_{j}} because b([Π_{𝔮_{j}}]_{C_{k,λ,n}}) = [𝔮_{j}]_{A_{k}} where [𝔮_{j}]_{A_{k}} is the class of 𝔮_{j} in A_{k}. We define a homomorphism b′ : A_{k} ⟶ C_{k,λ,n} by [𝔮_{j}]_{A_{k}} ↦ [Π_{𝔮_{j}}]_{C_{k,λ,n}} for all j = 1, ..., s. Clearly, b′ is a section of b, hence the above exact sequence splits. By class field theory, this implies that there is a cyclic extension k_{n}(λ)/k of degree p^{n}, which is linearly disjoint with k_{H}/k. From the construction, we know that λ is totally ramified in k_{n}(λ), and k_{n}(λ)/k is unramified outside λ. It is also clear that k_{n}(λ) is unique by class field theory.

As usual, we consider Kolyvagin’s derivative operator. Put G_{λ} = Gal(k_{n}(λ)/k), and fix a generator σ_{λ} of G_{λ} for λ ∈ P_{n}. We define N_{λ} = Σ_{i=0}^{p^{n}−1}σ_{λ}^{i} ∈ Z[G_{λ}] and D_{λ} = Σ_{i=0}^{p^{n}−1}iσ_{λ}^{i} ∈ Z[G_{λ}]. For a squarefree product 𝔞 = λ_{1} ·...·λ_{r} with λ_{i} ∈ P_{n}, we define k_{n}(𝔞) to be the compositum k_{n}(λ_{1})...k_{n}(λ_{r}), and K_{n,(𝔞)} = Kk_{n}(𝔞). We simply write K_{(𝔞)} for K_{n,(𝔞)} if no confusion arises. For 𝔞 = λ_{1} · ... · λ_{r}, we also define N_{𝔞} = ∏_{i=1}^{r} N_{λ_{i}} and D_{𝔞} = ∏_{i=1}^{r} D_{λ_{i}} ∈ Z[Gal(k_{n}(𝔞)/k)] = Z[Gal(K_{(𝔞)}/K)]. For a finite prime ρ of k which splits completely in K_{(𝔞)}, we take a prime ρ_{K_{(𝔞)}} of K_{(𝔞)}. By the standard method of Euler systems (cf. Lemmas 2.1 and 2.2 in Rubin [17], or Lemma 4.4.2 (i) in Rubin [18]), we know that there is a unique κ_{𝔞,ρ_{K_{(𝔞)}}} ∈ (K^{×} ⊗ Z/p^{n})^{∼} whose image in (K_{(𝔞)}^{×} ⊗ Z/p^{n})^{∼} coincides with D_{𝔞}(g_{K_{(𝔞)},ρ_{K_{(𝔞)}}} ). We also have an element δ_{𝔞} ∈ Z/p^{n}[Gal(K/k)]^{∼} such that D_{𝔞}θ_{K_{(𝔞)}}^{∼} ≡ ν_{K_{(𝔞)}/K}(δ_{𝔞}) (mod p^{n}) where ν_{K_{(𝔞)}/K} : Z_{p}[Gal(K/k)]^{∼} ⟶ Z_{p}[Gal(K_{(𝔞)}/k)]^{∼} is the map induced by σ ↦ ∑_{τ_{|_{K}} =σ} τ for σ ∈ Gal(K/k). This δ_{𝔞} is also determined uniquely by this property. We sometimes write κ_{𝔞} for κ_{𝔞,ρ_{K_{(𝔞)}}} if no confusion arises. We take an odd character χ of Gal(K/k) such that χ ≠ ω, and consider the χ-component κ_{𝔞}^{χ} ∈ (K^{×} ⊗ Z/p^{n})^{χ}, δ_{𝔞}^{χ} ∈ Z/p^{n}[Gal(K/k)]^{χ} = O_{χ}, ...etc.

Lemma 4.4. Put S_{i} = σ_{λ_{i}} − 1 ∈ O_{χ}[Gal(K_{(𝔞)}/K)]. Then, we have

θ_{K_{(𝔞)}}^{χ} ≡ (−1)^{r}δ_{𝔞}^{χ}S_{1} · ... · S_{r} (mod (p^{n}, S_{1}^{2}, ..., S_{r}^{2})).

Proof. We first prove θ_{K_{(𝔞)}}^{χ} ≡ aS_{1} · ... · S_{r} mod (S_{1}^{2}, ..., S_{r}^{2}) for some a ∈ O_{χ} by induction on r. For any subfields M_{1} and M_{2} such that K ⊂ M_{1} ⊂ M_{2} ⊂ K_{(𝔞)}, we denote by c_{M_{2}/M_{1}} : O_{χ}[Gal(M_{2}/K)] ⟶ O_{χ}[Gal(M_{1}/K)] the map induced by the restriction σ ↦ σ_{|_{M_{1}}} . Since c_{K_{(λ_{1})}/K}(θ_{K_{(λ_{1})}}^{χ}) = ((1−φ_{λ_{1}}^{−1})θ_{K})^{χ} (cf. Tate [23] p.86) and λ_{1} splits completely in K, we have c_{K_{(λ_{1})}/K}(θ_{K_{(λ_{1})}}^{χ}) = 0. Hence, S_{1} = σ_{λ_{1}} − 1 divides θ_{K_{(λ_{1})}}^{χ}. So the first assertion was verified for r = 1. Let 𝔞_{i} = 𝔞/λ_{i} for i with 1 ≤ i ≤ r. Then, we have c_{K_{(𝔞)}/K_{(𝔞_{i})}}(θ_{K_{(𝔞)}}^{χ}) = ((1 − φ_{λ_{i}}^{−1})θ_{K_{(𝔞_{i})}})^{χ}. Since λ_{i} splits completely in K, φ_{λ_{i}} is in Gal(K_{(𝔞_{i})}/K). Hence, 1 − φ_{λ_{i}}^{−1} is in the ideal I_{Gal(K_{(𝔞_{i})}/K)} = (S_{1}, ..., S_{i−1}, S_{i+1}, ..., S_{r}). This implies that c_{K_{(𝔞)}/K_{(𝔞_{i})}}(θ_{K_{(𝔞)}}^{χ}) is in the ideal (S_{1}^{2}, ..., S_{i−1}^{2}, S_{i+1}^{2}, ..., S_{r}^{2}) by the hypothesis of the induction. This holds for all i, so θ_{K_{(𝔞)}}^{χ} can be written as θ_{K_{(𝔞)}}^{χ} = α+β where α is divisible by all S_{i} for i = 1, ..., r, and β is in (S_{1}^{2}, ..., S_{r}^{2}). Therefore, θ_{K_{(𝔞)}}^{χ} ≡ aS_{1} · ... · S_{r} mod (S_{1}^{2}, ..., S_{r}^{2}) for some a ∈ O_{χ}. Next, we determine a mod p^{n}. Note that S_{i}D_{λ_{i}} ≡ −N_{λ_{i}} (mod p^{n}). Hence, S_{i}^{2}D_{λ_{i}} ≡ 0 (mod p^{n}). Thus, we have

D_{𝔞}(θ_{K_{(𝔞)}}^{χ}) ≡ D_{𝔞}(aS_{1} · ... · S_{r}) ≡ (−1)^{r}N_{𝔞}(a) (mod p^{n}).

Hence, N_{𝔞}((−1)^{r}a) = ν_{K_{(𝔞)}/K}((−1)^{r}a) ≡ ν_{K_{(𝔞)}/K}(δ_{𝔞}^{χ}) (mod p^{n}), which implies δ_{𝔞}^{χ} ≡ (−1)^{r}a (mod p^{n}) because ν_{K_{(𝔞)}/K} mod p^{n} is injective. This completes the proof of Lemma 4.4.

We put G = Gal(K_{(𝔞)}/K). As in §3, we have an isomorphism O_{χ}[G] ≃ A_{n,r} by the correspondence σ_{λ_{j}} ↔ 1 + S_{j} where A_{n,r} is the ring in 3.1 with O = O_{χ}. For i, s > 0 and θ_{K_{(𝔞)}}^{χ} ∈ O_{χ}[G], we have an ideal I_{i,s}(θ_{K_{(𝔞)}}^{χ}) of O_{χ} as in 3.2. By the definition of I_{i,s}(θ_{K_{(𝔞)}}^{χ}) and Lemma 4.4, we know that I_{r,1}(θ_{K_{(𝔞)}}^{χ}) is generated by δ_{𝔞}^{χ} and p^{n}. Thus, we get Corollary 4.5.

I_{r,1}(θ_{K_{(𝔞)}}^{χ}) = (δ_{𝔞}^{χ}, p^{n}).

For a prime λ of k, we define the subgroup Div_{K}^{λ} of Div_{K} ⊗Z_{p} by Div_{K}^{λ} = ⨁_{λ_{K}|λ} Z_{p}[λ_{K}] where λ_{K} ranges over all primes of K above λ. We fix a prime λ_{K}, then Div_{K}^{λ} = Z_{p}[Gal(K/k)/D_{λ_{K}}][λ_{K}] where D_{λ_{K}} is the decomposition group of λ_{K} in Gal(K/k). Let div_{λ} : (K^{×} ⊗ Z_{p})^{χ} ⟶ (Div_{K}^{λ})^{χ} be the map induced by the composite of div : K^{×} ⊗ Z_{p} ⟶ Div_{K} ⊗Z_{p} and the projection Div_{K} ⊗Z_{p} ⟶ Div_{K}^{λ}. The following lemma is immediate from the defining properties of κ_{𝔞,ρ_{K_{(𝔞)}}} and δ_{𝔞}, which we stated above.

Lemma 4.6. Assume that ρ is a finite prime of k which splits completely in K_{(𝔞)}. We take a prime ρ_{K_{(𝔞)}} of K_{(𝔞)} and a prime ρ_{K} of K such that ρ_{K_{(𝔞)}} | ρ_{K} | ρ.

(i) div_{ρ}(κ_{𝔞,ρ_{K_{(𝔞)}}}^{χ}) ≡ (δ_{𝔞}[ρ_{K}])^{χ} (mod p^{n}). (ii) If λ is prime to 𝔞ρ, we have div_{λ}(κ_{𝔞,ρ_{K_{(𝔞)}}}^{χ}) ≡ 0 (mod p^{n}).

We next proceed to an important property of κ_{𝔞,ρ_{K_{(𝔞)}}}^{χ}. Suppose that λ is a prime in P_{n} with (λ, 𝔞) = 1 and ρ is a prime with (ρ, 𝔞λ) = 1. We assume both ρ and λ split completely in K_{(𝔞)}. Put W = Ker(div_{λ} : (K^{×}⊗Z_{p})^{χ} ⟶ (Div_{K}^{λ})^{χ}), and R_{K}^{λ} = ⨁_{λ_{K}|λ} κ(λ_{K})^{×} where κ(λ_{K}) is the residue field of λ_{K} (κ(λ_{K}) coincides with the residue field κ(λ) = O_{k}/λ of λ because λ splits in K) and λ_{K} ranges over all primes of K above λ. We consider a natural map

ℓ_{λ} : W/W^{p^{n}} ⟶ (R_{K}^{λ}/(R_{K}^{λ})^{p^{n}})^{χ}

induced by x ↦ (x mod λ_{K}). Note that N(λ) ≡ 1 (mod p^{n}) because λ ∈ P_{n}. So, R_{K}^{λ}/(R_{K}^{λ})^{p^{n}} is a free Z/p^{n}Z[Gal(K/k)]-module of rank 1. We take a basis u ∈ (R_{K}^{λ}/(R_{K}^{λ})^{p^{n}})^{χ}, and define ℓ_{λ,u} : W/W^{p^{n}} ⟶ (Z/p^{n}Z[Gal(K/k)])^{χ} ≃ O_{χ}/(p^{n}) to be the composite of ℓ_{λ} and u ↦ 1. By Lemma 4.6 (ii), κ_{𝔞,ρ_{K_{(𝔞)}}}^{χ} is in W/W^{p^{n}} (note that W/W^{p^{n}} ⊂ (K^{×} ⊗ Z/p^{n}Z)^{χ}). We are interested in ℓ_{λ,u}(κ_{𝔞,ρ_{K_{(𝔞)}}}^{χ}). We take a prime ρ_{K_{(𝔞)}} (resp. λ_{K_{(𝔞)}}) of K_{(𝔞)} and a prime ρ_{K} (resp. λ_{K}) of K such that ρ_{K_{(𝔞)}} | ρ_{K} | ρ (resp. λ_{K_{(𝔞)}} | λ_{K} | λ).

Proposition 4.7. We assume that χ(𝔭) ≠ 1 for any prime 𝔭 of k above p, and that [ρ_{K}] and [λ_{K}] yield the same class in A_{K}^{χ}. Then, there is an element x ∈ W/W^{p^{n}} satisfying the following properties.

(i) For any prime λ′ of k such that (λ′, 𝔞) = 1, we have

div_{λ′}(κ_{𝔞,ρ_{K_{(𝔞)}}}^{χ} /x) ≡ 0 (mod p^{n}).

(ii) Choosing u suitably, we have

\frac{1 − N(λ)^{−1}}{p^{n}} ℓ_{λ,u}(x) ≡ δ_{𝔞λ}^{χ} (mod (δ_{𝔞}^{χ}, p^{n}))

where N(λ) = #κ(λ) = #(O_{k}/λ). In particular, in the case 𝔞 = (1) we can take x = g_{K,ρ_{K}}^{χ} mod W^{p^{n}}.

This proposition corresponds to Theorem 2.4 in Rubin [17], which was proved by using some extra property of the Gauss sums. For our g_{F,ρ_{F}} we do not have the property corresponding to Lemma 2.5 in [17], so we have to give here a proof in which we use only the definition of g_{F,ρ_{F}}, namely div(g_{F,ρ_{F}}^{χ}) = (θ_{F} [ρ_{F}])^{χ}.

Proof of Proposition 4.7. We denote by λ_{K_{(𝔞λ)}} the unique prime of K_{(𝔞λ)} above λ_{K_{(𝔞)}}. Put N = ord_{p}(N(λ) − 1) + 2n. We take by Chebotarev density theorem a prime ρ′ of k which splits completely in K_{(𝔞λ)}(μ_{p^{N}}) such that the class of [ρ′_{K_{(𝔞λ)}}] in A_{K_{(𝔞λ)}}^{χ} for a prime ρ′_{K_{(𝔞λ)}} of K_{(𝔞λ)} over ρ′ coincides with the class of [λ_{K_{(𝔞λ)}}]. Let ρ′_{K} be the prime below ρ′_{K_{(𝔞λ)}}. Then, the class of [λ_{K}], the class of [ρ′_{K}], and the class of [ρ_{K}] in A_{K}^{χ} all coincide. Hence, there is an element a ∈ W such that div(a) = [ρ_{K}]−[ρ′_{K}]. Define x ∈ W/W^{p^{n}} by x = κ_{𝔞,ρ′_{K_{(𝔞)}}}^{χ} ·a^{δ_{𝔞}^{χ}}.

By Lemma 4.6 (ii), div_{λ′}(κ_{𝔞,ρ_{K_{(𝔞)}}}^{χ} /x) ≡ 0 (mod p^{n}) for a prime λ′ such that (λ′, 𝔞ρρ′) = 1. By Lemma 4.6 (i), the same is true for λ′ = ρ and ρ′. Thus, we get the first assertion. In the case 𝔞 = (1), we take y = g_{K,ρ′_{K}}^{χ} a^{θ_{K}^{χ}}. Then, div(y) = div(g_{K,ρ_{K}}^{χ}), so y = g_{K,ρ_{K}}^{χ}, and we have g_{K,ρ_{K}}^{χ} mod W^{p^{n}} = y mod W^{p^{n}} = x. In order to show the second assertion, it is enough to prove

\frac{1 − N(λ)^{−1}}{p^{n}} ℓ_{λ,u}(κ_{𝔞,ρ′_{K_{(𝔞)}}}^{χ}) ≡ δ_{𝔞λ}^{χ} (mod p^{n})  (1)

for some u. Set Div_{K_{(𝔞λ)}}^{λ} = ⨁_{v|λ} Z_{p}[v] and R_{K_{(𝔞λ)}}^{λ} = ⨁_{v|λ} κ(v)^{×} = ⨁_{v|λ}(O_{K_{(𝔞λ)}}/v)^{×} where v ranges over all primes of K_{(𝔞λ)} above λ. Since the primes of K_{(𝔞)} above λ are totally ramified in K_{(𝔞λ)}, (Div_{K_{(𝔞λ)}}^{λ})^{χ} is isomorphic to O_{χ}[Gal(K_{(𝔞)}/K)] and (R_{K_{(𝔞λ)}}^{λ}/(R_{K_{(𝔞λ)}}^{λ})^{p^{n}})^{χ} is isomorphic to O_{χ}/(p^{n})[Gal(K_{(𝔞)}/K)]. We consider W_{K_{(𝔞λ)}} = Ker(div_{λ} : (K_{(𝔞λ)}^{×} ⊗ Z_{p})^{χ} ⟶ (Div_{K_{(𝔞λ)}}^{λ})^{χ}) and a natural map

ℓ_{λ,K_{(𝔞λ)}} : W_{K_{(𝔞λ)}}/W_{K_{(𝔞λ)}}^{p^{n}} ⟶ (R_{K_{(𝔞λ)}}^{λ}/(R_{K_{(𝔞λ)}}^{λ})^{p^{n}})^{χ}.

We take b ∈ (K_{(𝔞λ)}^{×} ⊗ Z_{p})^{χ} such that div(b) = [λ_{K_{(𝔞λ)}}] − [ρ′_{K_{(𝔞λ)}}]. Then, b′ = ℓ_{λ,K_{(𝔞λ)}}(b^{σ_{λ}−1}) is a generator of (R_{K_{(𝔞λ)}}^{λ}/(R_{K_{(𝔞λ)}}^{λ})^{p^{n}})^{χ} as an O_{χ}/(p^{n})[Gal(K_{(𝔞)}/K)]-module ([19] Chap.4 Prop.7 Cor.1). Using this b′, we identify (R_{K_{(𝔞λ)}}^{λ}/(R_{K_{(𝔞λ)}}^{λ})^{p^{n}})^{χ} with O_{χ}/(p^{n})[Gal(K_{(𝔞)}/K)], and define

ℓ_{λ,K_{(𝔞λ)},b′} : W_{K_{(𝔞λ)}}/W_{K_{(𝔞λ)}}^{p^{n}} ⟶ O_{χ}/p^{n}[Gal(K_{(𝔞)}/K)].

Since λ splits completely in K_{(𝔞)}, c_{K_{(𝔞λ)}/K_{(𝔞)}}(θ_{K_{(𝔞λ)}}^{χ}) = 0 by the formula in the proof of Lemma 4.1. Hence, σ_{λ} − 1 divides θ_{K_{(𝔞λ)}}^{χ}. Since (σ_{λ} − 1)[λ_{K_{(𝔞λ)}}] =

0, we have θ_{K_{(𝔞λ)}}^{χ}[λ_{K_{(𝔞λ)}}]^{χ} = 0. So, div(g_{K_{(𝔞λ)},ρ′_{K_{(𝔞λ)}}}^{χ}) = div((b^{−θ_{K_{(𝔞λ)}}} )^{χ}) = θ_{K_{(𝔞λ)}}^{χ}[ρ′_{K_{(𝔞λ)}}]^{χ}. The injectivity of div implies that g_{K_{(𝔞λ)},ρ′_{K_{(𝔞λ)}}}^{χ} = (b^{−θ_{K_{(𝔞λ)}}} )^{χ}. Further, by Lemma 4.4, we can write

θ_{K_{(𝔞λ)}}^{χ} ≡ (−1)^{r+1}δ_{𝔞λ}^{χ}S_{1} · ... · S_{r}(σ_{λ} − 1) + β (mod p^{n})

where β ∈ (S_{1}^{2}, ..., S_{r}^{2}, (σ_{λ} −1)^{2}). Since σ_{λ} −1 divides θ_{K_{(𝔞λ)}}^{χ}, σ_{λ} −1 also divides β. We write β = (σ_{λ} − 1)β′. So θ_{K_{(𝔞λ)}}^{χ} ≡ (σ_{λ} − 1)((−1)^{r+1}δ_{𝔞λ}^{χ}S_{1} · ... · S_{r} + β′) (mod p^{n}). Then,

ℓ_{λ,K_{(𝔞λ)},b′}(g_{K_{(𝔞λ)},ρ′_{K_{(𝔞λ)}}}^{χ}) = ℓ_{λ,K_{(𝔞λ)},b′}((b^{−θ_{K_{(𝔞λ)}}} )^{χ})
= −c_{K_{(𝔞λ)}/K_{(𝔞)}}((−1)^{r+1}δ_{𝔞λ}^{χ}S_{1} · ... · S_{r} + β′)
= (−1)^{r}δ_{𝔞λ}^{χ}S_{1} · ... · S_{r} − c_{K_{(𝔞λ)}/K_{(𝔞)}}(β′).

Since c_{K_{(𝔞λ)}/K_{(𝔞)}}(β′) ∈ (S_{1}^{2}, ..., S_{r}^{2}), using S_{i}D_{λ_{i}} ≡ −N_{λ_{i}} (mod p^{n}) and S_{i}^{2}D_{λ_{i}} ≡ 0 (mod p^{n}), we have

ℓ_{λ,K_{(𝔞λ)},b′}((g_{K_{(𝔞λ)},ρ′_{K_{(𝔞λ)}}}^{χ})^{D_{𝔞}}) = D_{𝔞}((−1)^{r}δ_{𝔞λ}^{χ}S_{1} · ... · S_{r} − c_{K_{(𝔞λ)}/K_{(𝔞)}}(β′))
= N_{𝔞}δ_{𝔞λ}^{χ}
= ν_{K_{(𝔞)}/K}(δ_{𝔞λ}^{χ}).

We similarly define W_{K_{(𝔞)}} = Ker(div_{λ} for K_{(𝔞)}) ⊂ (K_{(𝔞)}^{×} ⊗ Z_{p})^{χ}. Recall that W = Ker(div_{λ} for K) ⊂ (K^{×} ⊗ Z_{p})^{χ}. Let ℓ_{λ} (resp. ℓ_{λ,K_{(𝔞)}}, ℓ_{λ,K_{(𝔞λ)}}) be the natural map W_{K}/W_{K}^{p^{n}} ⟶ (R_{K}^{λ}/(R_{K}^{λ})^{p^{n}})^{χ} (resp. W_{K_{(𝔞)}}/W_{K_{(𝔞)}}^{p^{n}} ⟶ (R_{K_{(𝔞)}}^{λ}/(R_{K_{(𝔞)}}^{λ})^{p^{n}})^{χ}, W_{K_{(𝔞λ)}}/W_{K_{(𝔞λ)}}^{p^{n}} ⟶ (R_{K_{(𝔞λ)}}^{λ}/(R_{K_{(𝔞λ)}}^{λ})^{p^{n}})^{χ}). We have a commutative diagram

W_{K}/W_{K}^{p^{n}}  ⟶  W_{K_{(𝔞)}}/W_{K_{(𝔞)}}^{p^{n}}  ⟶  W_{K_{(𝔞λ)}}/W_{K_{(𝔞λ)}}^{p^{n}}
↓^{ℓ_{λ}}  ↓^{ℓ_{λ,K_{(𝔞)}}}  ↓^{ℓ_{λ,K_{(𝔞λ)}}}
(R_{K}^{λ}/(R_{K}^{λ})^{p^{n}})^{χ} ⟶ (R_{K_{(𝔞)}}^{λ}/(R_{K_{(𝔞)}}^{λ})^{p^{n}})^{χ} ⟶ (R_{K_{(𝔞λ)}}^{λ}/(R_{K_{(𝔞λ)}}^{λ})^{p^{n}})^{χ}

where the horizontal arrows are the natural maps. We take a generator u′ of (R_{K_{(𝔞)}}^{λ}/(R_{K_{(𝔞)}}^{λ})^{p^{n}})^{χ} as an O_{χ}/(p^{n})[Gal(K_{(𝔞)}/K)]-module, and a generator u′′ of (R_{K}^{λ}/(R_{K}^{λ})^{p^{n}})^{χ} as an O_{χ}/(p^{n})-module such that the diagram

W_{K}/W_{K}^{p^{n}} ⟶  W_{K_{(𝔞)}}/W_{K_{(𝔞)}}^{p^{n}}  ⟶  W_{K_{(𝔞λ)}}/W_{K_{(𝔞λ)}}^{p^{n}}
↓^{ℓ_{λ,u′′}}  ↓^{ℓ_{λ,K_{(𝔞)},u′}}  ↓^{ℓ_{λ,K_{(𝔞λ)},b′}}
O_{χ}/(p^{n}) ⟶^{ν_{K_{(𝔞)}/K}} O_{χ}/(p^{n})[Gal(K_{(𝔞)}/K)] ⟶^{id} O_{χ}/(p^{n})[Gal(K_{(𝔞)}/K)]

commutes where ν_{K_{(𝔞)}/K} is the norm map defined before Lemma 4.4, and id is the identity map.

Using the above computation of ℓ_{λ,K_{(𝔞λ)},b′}((g_{K_{(𝔞λ)},ρ′_{K_{(𝔞λ)}}}^{χ})^{D_{𝔞}}), if we get

\frac{1 − N(λ)^{−1}}{p^{n}} ℓ_{λ,K_{(𝔞)}}(g_{K_{(𝔞)},ρ′_{K_{(𝔞)}}}^{χ}) = ℓ_{λ,K_{(𝔞λ)}}(g_{K_{(𝔞λ)},ρ′_{K_{(𝔞λ)}}}^{χ}),  (2)

we obtain (1) from the above commutative diagram. The relation (2) is sometimes called the “congruence condition”, and can be proved by the method of Rubin [18] Corollary 4.8.1 and Kato [8] Prop.1.1. Put L = K_{(𝔞)}(μ_{p^{N}}) and L_{(λ)} = K_{(𝔞λ)}(μ_{p^{N}}) (Recall that N was chosen in the beginning of the proof). We take a prime ρ′_{L_{(λ)}} of L_{(λ)} above ρ′_{K_{(𝔞λ)}}, and

denote by ρ′_{L} the prime of L below ρ′_{L_{(λ)}}. We define ℓ_{λ,L} : W_{L}/W_{L}^{p^{N}} ⟶

(R_{L}^{λ}/(R_{L}^{λ})^{p^{N}})^{χ}, and ℓ_{λ,L_{(λ)}} : W_{L_{(λ)}}/W_{L_{(λ)}}^{p^{N}} ⟶ (R_{L_{(λ)}}^{λ}/(R_{L_{(λ)}}^{λ})^{p^{N}})^{χ} similarly.

We identify (R_{L}^{λ}/(R_{L}^{λ})^{p^{N}})^{χ} with (R_{L_{(λ)}}^{λ}/(R_{L_{(λ)}}^{λ})^{p^{N}})^{χ} by the map induced by the inclusion. Then, the norm map induces the multiplication by p^{n}. Since N_{L_{(λ)}/L}(g_{L_{(λ)},ρ′_{L_{(λ)}}}^{χ}) = (1 − φ_{λ}^{−1})g_{L,ρ′_{L}}^{χ}, we have p^{n}ℓ_{λ,L_{(λ)}}(g_{L_{(λ)},ρ′_{L_{(λ)}}}^{χ}) = (1 −

N(λ)^{−1})ℓ_{λ,L}(g_{L,ρ′_{L}}^{χ}). Hence,

ℓ_{λ,L_{(λ)}}(g_{L_{(λ)},ρ′_{L_{(λ)}}}^{χ}) ≡ p^{−n}(1 − N(λ)^{−1})ℓ_{λ,L}(g_{L,ρ′_{L}}^{χ}) (mod p^{N−n}).

Let S be the set of primes of k ramifying in L_{(λ)} and not ramifying in K_{(𝔞λ)}. Note that if 𝔭 ∈ S, 𝔭 is a prime above p. By Lemma 4.1 we have N_{L_{(λ)}/K_{(𝔞λ)}}(g_{L_{(λ)},ρ′_{L_{(λ)}}}^{χ}) = ϵ_{K_{(𝔞λ})}g_{K_{(𝔞λ)},ρ′_{K_{(𝔞λ)}}}^{χ} and N_{L/K_{(𝔞)}}(g_{L,ρ′_{L}}^{χ}) =

ϵ_{K_{(𝔞)}}g_{K_{(𝔞)},ρ′_{K_{(𝔞)}}}^{χ} where ϵ_{K_{(𝔞λ)}} = (Π_{𝔭∈S}(1 − φ_{𝔭}^{−1}))^{χ} ∈ O_{χ}[Gal(K_{(𝔞λ)}/K)] and

ϵ_{K_{(𝔞)}} = c_{K_{(𝔞λ)}/K_{(𝔞)}}(ϵ_{K_{(𝔞λ)}}) (c_{K_{(𝔞λ)}/K_{(𝔞)}} is the restriction map). Since we assumed χ(𝔭) ≠ 1 for all 𝔭 above p, ϵ_{K_{(𝔞λ)}} is a unit of O_{χ}[Gal(K_{(𝔞λ)}/K)]. Hence, we obtain (2) by taking the norms N_{L_{(λ)}/K_{(𝔞λ)}} of both sides of the above formula. This completes the proof of Proposition 4.7.

5 The other inclusion

In this section, for K and χ in Theorem 0.1 and i ≥ 0, we will prove Fitt_{i,O_{χ}}(A_{K}^{χ}) ⊂ (Θ_{i,K})^{χ} to complete the proof of Theorem 0.1. More precisely, we will show Fitt_{i,O_{χ}}(A_{K}^{χ}) ⊂ (Θ_{i,1,K})^{χ}. As in Theorem 0.1, suppose that

A_{K}^{χ} ≃ O_{χ}/(p^{n_{1}}) ⊕ ... ⊕ O_{χ}/(p^{n_{r}})

with 0 < n_{1} ≤ ... ≤ n_{r}. We take generators c_{1},...,c_{r} corresponding to the above isomorphism (c_{j} generates the j-th direct summand). Let P_{n} be as in §4. We define

Q_{j} = {λ ∈ P_{n} : there is a prime λ_{K} of K above λ such that
the class of λ_{K} in A_{K}^{χ} is c_{j}},

and Q = ⋃_{1≤j≤r} Q_{j}. We consider an exact sequence

0 ⟶ (K^{×} ⊗ Z_{p})^{χ} ⟶^{div} (Div_{K} ⊗Z_{p})^{χ} ⟶ A_{K}^{χ} ⟶ 0.

For λ ∈ Q, we have (Z_{p}[Gal(K/k)][λ_{K}])^{χ} = O_{χ}[λ_{K}]^{χ}. We define M_{Q} to be the inverse image of ⨁_{λ∈Q} O_{χ}[λ_{K}]^{χ} by div : (K^{×} ⊗ Z_{p})^{χ} ⟶ (Div_{K} ⊗Z_{p})^{χ}. On the other hand, as an abstract O_{χ}-module, A_{K}^{χ} fits into an exact sequence

0 ⟶ ⨁_{j=1}^{r} O_{χ}e_{j} ⟶^{f} ⨁_{j=1}^{r} O_{χ}e′_{j} ⟶^{g} A_{K}^{χ} ⟶ 0

where (e_{j}) and (e′_{j}) are bases of free O_{χ}-modules of rank r, f is the map e_{j} ↦ p^{n_{j}}e′_{j}, and g is induced by e′_{j} ↦ c_{j}. We define β : ⨁_{λ∈Q} O_{χ}[λ_{K}]^{χ} ⟶ ⨁_{j=1}^{r} O_{χ}e′_{j} by [λ_{K}]^{χ} ↦ e′_{j} for all λ ∈ Q_{j} and j = 1, ..., r. Then, β induces α : M_{Q} ⟶ ⨁_{j=1}^{r} O_{χ}e_{j}, and we have a commutative diagram of O_{χ}-modules

0 ⟶  M_{Q}  ⟶^{div} ⨁_{λ∈Q} O_{χ}[λ_{K}]^{χ} ⟶ A_{K}^{χ} ⟶ 0
↓^{α}  ↓^{β}  ∥
0 ⟶ ⨁_{j=1}^{r} O_{χ}e_{j} ⟶^{f} ⨁_{j=1}^{r} O_{χ}e′_{j}  ⟶^{g} A_{K}^{χ} ⟶ 0.

Put m = length_{O_{χ}}(A_{K}^{χ}). We take n > 0 such that n ≥ 2m and μ_{p^{n+1}} ⊄ K. We use the same notation as in Proposition 4.7. Especially, we consider

ℓ_{λ} : W/W^{p^{n}} ⟶ (R_{K}^{λ}/(R_{K}^{λ})^{p^{n}})^{χ} ≃ O_{χ}/(p^{n})

for λ ∈ Q.

Lemma 5.1. Suppose that 𝔞, λ, ρ,...etc satisfy the hypotheses of Proposition 4.7. We further assume that the primes dividing 𝔞λρ are all in Q. Then, there exists κ̃_{𝔞,ρ_{K_{(𝔞)}}}^{χ} ∈ M_{Q} satisfying the following properties.

(i) For any prime λ′ such that (λ′, 𝔞ρ) = 1, we have

div_{λ′}(κ̃_{𝔞,ρ_{K_{(𝔞)}}}^{χ}) = 0.

(ii)

div_{ρ}(κ̃_{𝔞,ρ_{K_{(𝔞)}}}^{χ}) ≡ (δ_{𝔞}[ρ_{K}])^{χ} (mod p^{n}).

(iii)

\frac{1 − N(λ)^{−1}}{p^{n}} ℓ_{λ}(κ̃_{𝔞,ρ_{K_{(𝔞)}}}^{χ}) ≡ uδ_{𝔞λ}^{χ} (mod (δ_{𝔞}^{χ}, p^{m}))

for some u ∈ O_{χ}^{×}.

Proof. Let x be an element in W/W^{p^{n}}, which satisfies the conditions in Proposition 4.7, and take a lifting y ∈ W of x. By Proposition 4.7 (i) and Lemma 4.6, we can write div(y) = A + p^{n}B where A is a divisor whose support is contained in the primes dividing 𝔞ρ. Since the class of p^{m}B in A_{K}^{χ} is zero, we can take z ∈ (K^{×} ⊗ Z_{p})^{χ} such that div(z) = p^{m}B. Put κ̃_{𝔞,ρ_{K_{(𝔞)}}}^{χ} = y/z^{p^{n−m}}. Then, κ̃_{𝔞,ρ_{K_{(𝔞)}}}^{χ} is in M_{Q}, and satisfies the above properties (i), (ii), and (iii) by Proposition 4.7 and Lemma 4.6.

We will prove Theorem 0.1. First of all, as we saw in Proposition 3.2, Fitt_{0,O_{χ}}(A_{K}^{χ}) = Θ_{0,K}^{χ}. Recall that we put m = length_{O_{χ}}(A_{K}^{χ}), so Fitt_{0,O_{χ}}(A_{K}^{χ}) = (p^{m}). Next, we consider the commutative diagram before Lemma 5.1. We denote by α_{j} : M_{Q} ⟶ O_{χ}e_{j} ≃ O_{χ} the composite of α and the j-th projection. We take ρ_{r} ∈ Q_{r} and a prime ρ_{r,K} of K above ρ_{r}. We consider g_{K,ρ_{r,K}}^{χ} ∈ M_{Q}. We choose λ_{r} ∈ Q_{r} such that λ_{r} ≠ ρ_{r}, ord_{p}(N(λ_{r}) − 1) = n, the class of λ_{r,K} in A_{K}^{χ} coincides with the class of ρ_{r,K}, and α_{r}(g_{K,ρ_{r,K}}^{χ}) mod p^{n} = u′ℓ_{λ_{r}}(g_{K,ρ_{r,K}}^{χ}) for some u′ ∈ O_{χ}^{×}. This is possible by Chebotarev density theorem (Theorem 3.1 in Rubin [17], cf. also [16]). By the commutative diagram before Lemma 5.1 and div(g_{K,ρ_{r,K}}^{χ}) = θ_{K}^{χ}[ρ_{r,K}]^{χ}, we have

ord_{p}(α_{r}(g_{K,ρ_{r,K}}^{χ})) + n_{r} = ord_{p}(θ_{K}^{χ}) = m.  (3)

On the other hand, by Proposition 4.7, we have ℓ_{λ_{r}}(g_{K,ρ_{r,K}}^{χ}) = uδ_{λ_{r}}^{χ} mod (p^{m}) for some u ∈ O_{χ}^{×}. Hence, α_{r}(g_{K,ρ_{r,K}}^{χ}) ≡ u′ℓ_{λ_{r}}(g_{K,ρ_{r,K}}^{χ}) ≡ uu′δ_{λ_{r}}^{χ} mod (p^{m}). From (3), δ_{λ_{r}}^{χ} mod p^{m} ≠ 0, hence, ord_{p}(α_{r}(g_{K,ρ_{r,K}}^{χ})) = ord_{p}(δ_{λ_{r}}^{χ}) (for a nonzero element x in O_{χ}/p^{m}, ord_{p}(x) is defined to be ord_{p}(x̃) where x̃ is a lifting of x to O_{χ}). Therefore, we have

ord_{p}(δ_{λ_{r}}^{χ}) = m − n_{r}.

Hence, Fitt_{1,O_{χ}}(A_{K}^{χ}) = (p^{m−n_{r}}) is generated by I_{1,1}(θ_{K_{(λr)}}^{χ}) by Corollary 4.5. Thus, Fitt_{1,O_{χ}}(A_{K}^{χ}) ⊂ (Θ_{1,1,K})^{χ} ⊂ (Θ_{1,K})^{χ}. For any i > 1, we prove Fitt_{i,O_{χ}}(A_{K}^{χ}) ⊂ Θ_{i,K}^{χ} by the same method. We will show that we can take λ_{r} ∈ Q_{r}, λ_{r−1} ∈ Q_{r−1},...inductively such that δ_{𝔞_{i}}^{χ} generates Fitt_{i,O_{χ}}(A_{K}^{χ}) where 𝔞_{i} = λ_{r} · ... · λ_{r−i+1}. For i such that 1 < i ≤ r, suppose that λ_{r},...,λ_{r−i+2} were defined. We first take ρ_{r−i+1} ∈ Q_{r−i+1}, which splits completely in K_{(𝔞_{i−1})}. We consider κ = κ̃_{𝔞_{i−1},ρ_{r−i+1,K_{(𝔞_{i−1})}}}^{χ} ∈ M_{Q} where we used the notation in Lemma 5.1. We choose λ_{r−i+1} ∈ Q_{r−i+1} such that λ_{r−i+1} ≠ ρ_{r−i+1}, ord_{p}(N(λ_{r−i+1})−1) = n, λ_{r−i+1} splits completely in K_{(𝔞_{i−1})}, the class of λ_{r−i+1,K} in A_{K}^{χ} coincides with the class of ρ_{r−i+1,K} in A_{K}^{χ}, and α_{r−i+1}(κ) mod p^{n} = u′ℓ_{λ_{r−i+1}}(κ) for some u′ ∈ O_{χ}^{×}. This is also possible by Chebotarev density theorem (Theorem 3.1 in Rubin [17], cf. also [16]). By Lemma 5.1 (ii), div_{ρ_{r−i+1}}(κ) ≡ δ_{𝔞_{i−1}}^{χ}[ρ_{r−i+1}]^{χ} (mod p^{n}). Hence, from the commutative diagram before Lemma 5.1, we obtain

ord_{p}(α_{r−i+1}(κ)) + n_{r−i+1} = ord_{p}(δ_{𝔞_{i−1}}^{χ}).

By the hypothesis of the induction, we have ord_{p}(δ_{𝔞_{i−1}}^{χ}) = n_{1} + ... + n_{r−i+1}. It follows that

ord_{p}(α_{r−i+1}(κ)) = n_{1} + ... + n_{r−i}.

On the other hand, by Lemma 5.1 (iii), we have

ord_{p}(α_{r−i+1}(κ)) = ord_{p}(ℓ_{λ_{r−i+1}}(κ)) = ord_{p}(δ_{𝔞_{i−1}λ_{r−i+1}}^{χ})
= ord_{p}(δ_{𝔞_{i}}^{χ}).

Therefore,

ord_{p}(δ_{𝔞_{i}}^{χ}) = n_{1} + ... + n_{r−i}.

This shows that δ_{𝔞_{i}}^{χ} generates Fitt_{i,O_{χ}}(A_{K}^{χ}) = (p^{n_{1}+...+n_{r−i}}). Hence, by Corollary 4.5 we obtain

I_{i,1}(θ_{K_{(𝔞_{i})}}^{χ}) = (δ_{𝔞_{i}}^{χ}) = Fitt_{i,O_{χ}}(A_{K}^{χ}).

Thus, we have Fitt_{i,O_{χ}}(A_{K}^{χ}) ⊂ (Θ_{i,1,K})^{χ} ⊂ (Θ_{i,K})^{χ}. Note that for i = r, we have got Θ_{r,K}^{χ} = (1). Hence, Θ_{i,K}^{χ} = (1) for all i ≥ r, and we have Fitt_{i,O_{χ}}(A_{K}^{χ}) = Θ_{i,K}^{χ} for all i ≥ 0. This completes the proof of Theorem 0.1.

A appendix

In this appendix, we determine the initial Fitting ideal of the Pontrjagin dual (A_{F_{∞}}^{∼})^{∨} (cf. §2) of the non-ω component of the p-primary component of the ideal class group as a Z_{p}[[Gal(F_{∞}/k)]]-module for the cyclotomic Z_{p}-extension F_{∞} of a CM-field F such that F/k is finite and abelian, under the assumption that the Leopoldt conjecture holds for k and the μ-invariant of F vanishes. Our aim is to prove Theorem A.5. For the initial Fitting ideal of the Iwasawa module X_{F_{∞}} = lim_{←}A_{F_{n}} of F_{∞}, see [11] and Greither’s results [3], [4]. Suppose that λ_{1},...,λ_{r} are all finite primes of k, which are prime to p and ramifying in F_{∞}/k. We denote by P_{λ_{i}} the p-Sylow subgroup of the inertia subgroup of λ_{i} in Gal(F_{∞}/k). We first assume that

(∗)  P_{λ_{1}} × ... × P_{λ_{r}} ⊂ Gal(F_{∞}/k).

(Compare this condition with the condition (A_{p}) in [11] §3.) We define a set H of certain subgroups of Gal(F_{∞}/k) by

H = {H_{1}×...×H_{r} | H_{i} is a subgroup of P_{λ_{i}} for all i such that 1 ≤ i ≤ r}.

We also define

M = {M_{∞} | k ⊂ M_{∞} ⊂ F_{∞}, M is the fixed field of some H ∈ H}.

For an intermediate field M_{∞} of F_{∞}/k, we denote by

ν_{F_{∞}/M_{∞}} : Z_{p}[[Gal(M_{∞}/k)]]^{∼} ⟶ Z_{p}[[Gal(F_{∞}/k)]]^{∼}

the map induced by σ ↦ Σ_{τ_{|_{M∞}} =σ}τ for σ ∈ Gal(M_{∞}/k). We define Θ_{F_{∞}/k}^{∼} to be the Z_{p}[[Gal(F_{∞}/k)]]^{∼}-module generated by ν_{F_{∞}/M_{∞}}(θ_{M_{∞}}^{∼}) for all M_{∞} ∈ M_{F_{∞}/k}. Put Λ_{F} = Z_{p}[[Gal(F_{∞}/k)]]. Let ι : Λ_{F} ⟶ Λ_{F} be the map defined by σ ↦ σ^{−1} for all σ ∈ Gal(F_{∞}/k). For a Λ_{F} -module M, we denote by M^{≈} to be the component obtained from M^{−} by removing M^{ω^{−1}}, namely M^{−} = M^{≈} ⊕M^{ω^{−1}} if μ_{p} ⊂ F, and M^{−} = M^{≈} otherwise (cf. 1.1). The map ι induces M^{∼} ⟶^{ι} M^{≈} which is bijective.

Proposition A.1. We assume that the μ-invariant of F vanishes. Under the assumption of (∗), we have

Fitt_{0,Λ_{F}} ((A_{F_{∞}}^{∼})^{∨})^{≈} = ι(Θ_{F_{∞}/k}^{∼}).

Proof. This can be proved by the same method as the proof of Theorem 0.9 in [11] by using a slight modification of Lemma 4.1 in [11]. In fact, instead of Corollary 5.3 in [11], we can use

Lemma A.2. Let L/K be a finite abelian p-extension of CM-fields. Suppose that P is a set of primes of K_{∞} which are ramified in L_{∞} and prime to p. For v ∈ P, e_{v} denotes the ramification index of v in L_{∞}/K_{∞}. Then, we have an exact sequence

0 ⟶ A_{K_{∞}}^{∼} ⟶ (A_{L_{∞}}^{∼})^{Gal(L_{∞}/K_{∞})} ⟶ (⨁_{v∈P} Z/e_{v}Z)^{∼} ⟶ 0

Proof of Lemma A.2. It is enough to prove Ĥ^{0}(L_{∞}/K_{∞}, A_{L_{∞}}^{∼}) = (⨁_{v∈P} Z/e_{v}Z)^{∼}. Let P′_{n} be the set of primes of K_{n} ramifying in L_{n}. Then, by Lemma 5.1 (ii) in [11], we have Ĥ^{0}(L_{n}/K_{n}, A_{L_{n}}^{∼}) = (⨁_{v∈P′_{n}} H^{1}(L_{n,w}/K_{n,v}, O_{L_{n,w}}^{×}))^{∼} = (⨁_{v∈P′_{n}} Z/e_{v}Z)^{∼} where w is a prime of L_{n} above v. If v is a prime above p, it is totally ramified in K_{∞} for sufficiently large n, hence we have lim_{→}(⨁_{v∈P′_{n},v|p} Z/e_{v}Z)^{∼} = 0. On the other hand, lim_{→}(⨁_{v∈P′_{n},v |̸p} Z/e_{v}Z)^{∼} = (⨁_{v∈P} Z/e_{v}Z)^{∼}. Thus, we get

Ĥ^{0}(L_{∞}/K_{∞}, A_{L_{∞}}^{∼}) = (⨁_{v∈P} Z/e_{v}Z)^{∼}.

Next, we consider a general CM-field F with F/k finite and abelian. We assume that the Leopodt conjecture holds for k.

Lemma A.3. (Iwasawa) Let λ be a prime of k not lying above p. Suppose that k(λp^{∞}) is the maximal abelian pro-p extension of k, unramified outside pλ. Then the ramification index of λ in k(λp^{∞}) is p^{n_{λ}} where n_{λ} = ord_{p}(N(λ) − 1).

In fact, Iwasawa proved that the Leopoldt conjecture implies the existence of “λ-field” (𝔮-field) in his terminology ([5] Theorem 1). This means that the ramification index of λ is p^{n_{λ}}.

Lemma A.4. Let F/k be a finite abelian extension such that F is a CM-field. Then, there is an abelian extension F′/k satisfying the following properties.

(i) F′_{∞} ⊃ F_{∞}, and the extension F′_{∞}/F_{∞} is a finite abelian p-extension which is unramified outside p. (ii) F′_{∞} satisfies the condition (∗).

Proof. This follows from Lemme 2.2 (ii) in Gras [2], but we will give here a proof. Suppose that λ_{1},...,λ_{r} are all finite primes ramifying in F_{∞}/k, and prime to p. We denote by e_{λ_{i}}^{(p)} the p-component of the ramification index of λ_{i} in F_{∞}. By class field theory, e_{λ_{i}}^{(p)} ≤ p^{n_{λ_{i}}}. We take a subfield k_{i} of k(λ_{i}p^{∞}) such that k_{i}/k is a p-extension, and the ramification index of k_{i}/k is e_{λ_{i}}^{(p)}. This is possible by Lemma A.3. Take F′ such that F′_{∞} = F_{∞}k_{1}...k_{r}. It is clear that F′ satisfies the condition (i). Since k_{1}...k_{r} ⊂ F′_{∞}, F′ satisfies the condition (∗).

We define ι(Θ_{F_{∞}/k}^{∼}) by ι(Θ_{F_{∞}/k}^{∼}) = c_{F′_{∞}/F_{∞}}(ι(Θ_{F′_{∞}/k}^{∼})) where c_{F′_{∞}/F_{∞}} : Λ_{F′} ⟶ Λ_{F} is the restriction map. This ι(Θ_{F_{∞}/k}^{∼}) does not depend on the choice of F′. In fact, we have

Theorem A.5. We assume the Leopoldt conjecture for k and the vanishing of the μ-invariant of F. Then, we have

Fitt_{0,Λ_{F}} ((A_{F_{∞}}^{∼})^{∨})^{≈} = ι(Θ_{F_{∞}/k}^{∼}).

Proof. We take F′ as in Lemma A.4. By Proposition A.1, Theorem A.5 holds for F′_{∞}. Since F′_{∞}/F_{∞} is unramified outside p, by Lemma A.2 the natural map A_{F_{∞}}^{∼} ⟶ (A_{F′_{∞}}^{∼})^{Gal(F′_{∞}/F_{∞})} is an isomorphism. Hence, we get

Fitt_{0,Λ_{F}} ((A_{F_{∞}}^{∼})^{∨})^{≈} = c_{F′_{∞}/F_{∞}}(Fitt_{0,Λ_{F′}} ((A_{F′_{∞}}^{∼})^{∨})^{≈}) = c_{F′_{∞}/F_{∞}}(ι(Θ_{F′_{∞}/k}^{∼}))
= ι(Θ_{F_{∞}/k}^{∼}).

This completes the proof of Theorem A.5.

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Masato Kurihara

Department of Mathematics Tokyo Metropolitan University Hachioji, Tokyo, 192-0397 Japan kurihara@math.metro-u.ac.jp 564

Documenta Mathematica · Extra Volume Kato (2003)